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In this paper we study the existence and uniqueness of the strong solution offollowing d dimensional stochastic differential equation (SDE) driven byBrownian motion: dX(t)=b(t,X(t))dt+a(t,X(t))dB(t), X(0)= x, where B is ad-dimensional standard Brownian motion; the diffusion coefficient a is a Holdercontinuous and uniformly non-degenerate matrix-valued function and the driftcoefficient b may be discontinuous and unbounded, not necessarily in Sobolevspace, extending the previous works to discontinuous and unbounded driftcoefficient situation. The idea is to combine the Zvonkin transformation withthe Lyapunov function approach. To this end, we need to establish a localversion of the connection between the solutions of the SDE up to the exit timeof a bounded connected open set D and the associated partial differentialequation on this domain. As an interesting byproduct, we establish a localizedversion of the Krylov estimates (Theorem 4.1) and a localized version of thestability result of the stochastic differential equations of discontinuouscoefficients (Theorem 4.5).

Strong solution of stochastic differential equations with discontinuous and unbounded coefficients